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Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product).

Assume $A,B\in\bigotimes_{k=1}^p\mathbb R^{n_k}$ with$^1$ $$A=C^{(1)}\ast\cdots\ast C^{(p)}\tag1$$ and $$B=D^{(1)}\ast\cdots\ast D^{(p)}\tag2,$$ where $\ast$ denotes the tensor contraction with respect to the last and first mode of the left and right operatand, respectively, $C^{(1)}\in\mathbb R^{n_1\times r_1}$, $C^{(k)}\in\mathbb R^{r_{k-1}\times n_k\times r_k}$ for $k\in\{2,\ldots,p-1\}$, $C^{(p)}\in\mathbb R^{r_{p-1}\times n_p}$ and $D^{(1)}\in\mathbb R^{n_1\times s_1}$, $D^{(k)}\in\mathbb R^{s_{k-1}\times n_k\times s_k}$ for $k\in\{2,\ldots,p-1\}$, $D^{(p)}\in\mathbb R^{s_{p-1}\times n_p}$.

Are we able to express the inner product $\langle A,B\rangle$ in terms of tensor contractions?


$^1$ If $U\in\mathbb R^{n_1\times\cdots\times n_p\times r}$ and $V\in\mathbb R^{r\times m_1\times\cdots\times m_q}$, then $$(U\ast V)_{i_1\:,\:\ldots\:,\:i_p,\:j_1,\:\ldots\:,\:j_q}:=\sum_{k=1}^rU_{i_1\:,\:\ldots\:,\:i_p,\:r}V_{r,\:j_1,\:\ldots\:,\:j_q}.$$

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